Bernouilli's problem in the Acta Eruditorum for October 1698

Author: Isaac Newton

Source: MS Add. 3968, ff. 369r-371v, Cambridge University Library, Cambridge, UK

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<369r>

In Actis Eruditorum pro mense Octobri Anni 1698, pag. 471, D. Iohannes Bernoullius hac scripsit. {illeg} Methodum quam optaveram generalem secandi [curvas] ordinatim positione datas sive algebraicas sive trascendentales in angulo recto sive obliquo, invariabili sive data Lege variabili, tandem ex voto erui, cui Leibnitio approbatore, ne γρυ addi posset ad ulteriorem perfectionem, et vel ideo tantum quod perpetuo ad æquationem deducat in qua si interdum indeterminatæ sunt inseparabiles, Methodus non ideo imperfectior est, Non enim hujus sed alius est Methodi indeterminatas separare Rogamus igitur fratrem ut velit suas quoque vires exercere in re tanti Momenti. Suscepti Laboris Non pœnitebit si felix successus fructu jucundo compensaverit. Scio {r}elicturum suum quem nunc fovet modum, qui in paucissimis tantum exemplis adhiberi potest.

Hi tres viri celeberrimi sese jam ab annis quatuor vel quinque circiter in solvendis hujusmodi Problematibus exercuerant. Absque spiritu divinandi eandem solutionem cum Bernoulliana tradere difficile fuerit. Sufficit quod solutio sequens sit generalis, et ad æquationem semper deducat.

Problema

Quæritur Methodus generalis inveniendi seriem Curvarum quæ Curvat in serie alia quacumque data constitutas, ad angulum vel datum, vel data Lege variabilem secabunt.

Solutio.

Natura Curvarum secandarum dat tangentes earundem <369v> ad intersectionum puncta quæcumque, et anguli intersectionum dant perpendicula Curvarum secantium, et perpendicula duo coeuntia, per concursum suum ultimum, dant centrum curvaminis Curvæ secantis ad punetum intersectionis cujuscumque. Ducatur Abscissa in situ quocumque commodo, et sit ejus fluxio unitas, et positio perpendiculi dabit fluxionem primam Ordinatæ ad Curvam quæsitam pertinentis, et curvamen hujus Curvæ dabit fluxionem secundam ejusdem Ordinatæ. Et sic Problema semper deducetur ad æquationes. Quod erat faciendum.

Scholium.

Non hujus, sed alius est methodi æquationes reducere, et indeterminabus separare. \ad æqu |in| series convergentes, ubi opus est deducere convertere. Nam solutio Problematis per Newtoni Analysin universalem general|ior| \det/ evadet./ Problema hocce, cum Nullius fere sit usus, in Actis Eruditorum annos plures Neglectum, et insolutum Mansit. Et eadem de causa solutionem ejus Non ulterius prosequor.

<371r>

If a \given/ series of Curves drawn of the same kind succeeding one another in an uniform manner according to any general rule {b}e {illeg} \one of w^ch is BD/ are to be cut by another curve CD in given angle are in right or oblique invariable or variable by any given Rule: let D be any point of intersection & the \nature of the intersected Curves & the {illeg} two Rules/ Rules will give the angle of intersection, the perpendicular to the Curve desired & the center of its Curvity. Let the Ordinate of the Curve desired be rep \And/ The perpendicular will give the first fluxion of the Ordinate & the curvity will give the second fluxion. \2/Let the Ordinate be represented by the area of another \Curve/ upon y^e same abscissa, & the second fl first fluxion will be represented by the Ordinate of this other curve & the second fluxion by the proportion of the \Ordinate to the/ subtangent|.| to the Ordinate. \1/And so the Probleme is reduced to {illeg} equations involving first & second fluxions. Let 3 and so the Probleme is reduced to \the c{illeg} area ordin{illeg} the/ property of the tangent of a curve.

S^r

M^r Iohn Bernoulli in the Acta Eruditorum for October 1698. pag. 471 wri|o|tes thus \in this manner./. {sic} Methodum quam optaveram generalem secandi \[curvas]/ ordinatim positione datas sive algebraicas sive mechanicas trascendentes, in angulo recto, sive obliquo, invariabili, sive dat{illeg}|a| lege variabili, tandem ex voto erui, cui Leibnitio approbatore, ne γρυ addi posset ad ulteriorem perfectionem, et vel ideo tantum quod perpetuo ad æquationem deducat, in qua si interdum indeterminatæ sunt inseparabiles, methodus non ideo imperfectior est, non enim hujus sed alius est methodi indeterminatas separare. Rogamus it fratrem ut velit suas quo vires \exercere/ in re tanti momenti. Suscepti laboris non penitebit si fœlix successus fructu jucundo compensaverit. {illeg} To give the {illeg} |Scio relicturum suum quem nunc fovet modum qui in paucissimus tantum exemplis adhibere potest.| These Gentlemen had been four or five years about Problemes of this kind, & to give the very same solution {illeg} with that here mentioned might require a spirit of divination. But the Probleme may be generally solved after the following ma{illeg}|n|ner.

The Probleme

If a|A| series of Curves of \being given of one &/ the same kind, succeeding one another (in forme & position) in an uniform manner according to any general Rule are to be cut it \being/ given & \find/ another Curve is to be found w^ch shall cut |all| the Curves in the said Series, in any angle right, or oblique, invariable, or variable according to any Rule given \assigned/ Rule assigned.

The method of Solution

Let BD be \any/ one of the Curves in the Series, CD the Curve w^ch is to cut it, D the point of intersection \&/ AE the \common/ Abscissa \of the two Curves/ & ED the \common/ ordinate of the \two/ Curves: & the two Rules will \give/ the \[the/ angle of intersection \at D the point D/] the perpendicular to the Curve CD & the center \radius & radius/ of its curvity at the point D, & the position of the perpendicular will \give/ the first fluxion & the {illeg} curvity the second fluxion of it the ordinate of the \same/ curve D. And so the Probleme is \will be/ reduced to equations involving fluxions And how to manage these Equations is not the business of this Probleme \& by Separating or extracting/ the fluents will be resolved.

M^r Leibnitz in the Acta Eruditorum for Ma

When M^r Fatio asserted the method of fluxions to M^r Newton M^r Leibnits challenged him to solve this Problem in the Acta Eruditorum for May 1700 pag 204, challenged M^r Fatio to solve this Probleme, Invenire Curvam aut saltem proprietatem tangentium Curvæ quæ Curvas etiam transcendentes ordinatim datas secet ad angulos rectos. Let ED the Ordinate of the Curve CD be represented by the area of another Curve upon the same Abscissa CD AE & the first fluxion of this Ordinate will be represented by the Ordinate of this|e| other Curve & the second fluxion by the proportion of this last Ordinate to y^e subtangent of the other curve. And so the property of the Tangent of the other Curve is given.

Let the Ordinate of the Curve desired be represented by the area of <371v> another curve of \upon/ the same Abscissa & the first fluxions will be represented by this|e| Ordinate of the|is| other Curve, & the second fluxion by the proportion of this|e| Ordinate to the subtangent|.| &|A|nd so the Probleme is reduced to the {illeg} property of {all} of the Tangent of this other Curve & to the Quadrature thereof \of a Tangent/

and the first Rule will give the tangent of the Curve BD at the point D, & the second Rule will give the angle of intersection & tangent of the other Curve CD at y^e same point D, & both the Rules together will give the Radius of its curvity & the fluxion of & both the Rules together {illeg} will give the Radius of the curvity of the other Curve CD at the same point D. Let the Abscissa \AE/ flow uniformly & its fluxion be called 1, & the position of the Tangent of y^e Curve CD will give the first fluxion of th its Ordinate ED, & the Curvity of the same Curve CD at the point D will give the second fluxion of the same Ordinate And so the Problem will be reduced to equations involving the first & second fluxions of the Ordinate of the Curve desired, & by \reducing the Equations &/ extracting or separt|a|ting the fluents will (w^ch is not the business of this Probleme \method/) will be resolved.

There may be some Art in chusing the Ordinat Abscissa & Ordinate or other Fluents to which the invention of y^e Curve CD may be best referred \& in reducing the Equations after the best manner./. {sic} But nothing more is here desired then a general method of resolving the Probleme without entring into particular cases.

The curvity of the \intersecting/ Curve CD at y^e point D is found by taking in the tangent of \that intersecting Curve/ CD another point d infinitely near to y^e point D, \&/ finding the tangent at y^e point d of the Curve in the series w^ch passes through that point, \d/ & \also/ the tangent of the intersecting Curve \intersecting it/ at y^e same point d; & at upon the \two/ intersecting curves at y^e points D & d of intersection D & d erecting perpendiculars. For these perpendiculars shall intersect one another at y^e Center of \the/ curvity of the intersecting Curves.